Critical issues with the Pearson's chi-square test

Pearson's chi-square tests are among the most commonly applied statistical tools across a wide range of scientific disciplines, including medicine, engineering, biology, sociology, marketing and business. However, its usage in some areas is not correct. For example, the chi-square test for homogeneity of proportions (that is, comparing proportions across groups in a contingency table) is frequently used to verify if the rows of a given nonnegative \ (m n\) (contingency) matrix \ (A\) are proportional. The null hypothesis \ (H₀\): ``\ (m\) rows are proportional'' (for the whole population) is rejected with confidence level \ (1 - \) if and only if \ (²ₒₓ₀ₓ > ²₂ₑ₈ₓ\), where the first term is given by Pearson's formula, while the second one depends only on \ (m, n\), and \ (, \) but not on the entries of \ (A\). It is immediate to notice that the Pearson's formula is not invariant. More precisely, whenever we multiply all entries of \ (A\) by a constant \ (c\), the value \ (²ₒₓ₀ₓ (A) \) is multiplied by \ (c\), too, \ (²ₒₓ₀ₓ (cA) = c ²ₒₓ₀ₓ (A) \). Thus, if all rows of \ (A\) are exactly proportional then \ (²ₒₓ₀ₓ (cA) = c ²ₒₓ₀ₓ (A) = 0\) for any \ (c\) and any \ (\). Otherwise, \ (²ₒₓ₀ₓ (cA) \) becomes arbitrary large or small, as positive \ (c\) is increasing or decreasing. Hence, at any fixed significance level \ (\), the null hypothesis \ (H₀\) will be rejected with confidence \ (1 - \), when \ (c\) is sufficiently large and not rejected when \ (c\) is sufficiently small. Yet, obviously, the rows of \ (cA\) should be proportional or not for all \ (c\) simultaneously. For this reason, Pearson's test certainly cannot be applied to ``physical data'', which are obtained by measurements. Indeed, in this case matrix \ (A\) depends on the unit of measurement. The test can be applied only to categorical data and even then some further limitations are required, which we consider in this paper.